On the algebraic and arithmetic structure of the monoid of product-one sequences

Abstract

Let $G$ be a finite group. A finite unordered sequence $S=g_1\bullet \cdots \bullet g_{\ell }$ of terms from $G$, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals $1_G$, the identity element of the group. As usual, we consider sequences as elements of the free abelian monoid $\mathcal{ℱ}\left(G\right)$ with basis $G$, and we study the submonoid $\mathcal{ℬ}\left(G\right)\subset \mathcal{ℱ}\left(G\right)$ of all product-one sequences. This is a finitely generated C-monoid, which is a Krull monoid if and only if $G$ is abelian. In case of abelian groups, $\mathcal{ℬ}\left(G\right)$ is a well-studied object. In the present paper we focus on nonabelian groups, and we study the class semigroup and the arithmetic of $\mathcal{ℬ}\left(G\right)$.